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  "body": "\n## Description\n\nA tree, in set theory, is a [[partially ordered set|011C29Q1S1]] $(T, \u003c)$ in which, for each $t \\in T$, the set $\\set{s \\in T : s \u003c t}$ is [[well-ordered|0F1D6PK9RZ]].\nThe height of $t$ is the order type of this set, and level $\\alpha$ of $T$ is the set of elements of height $\\alpha$.\nA branch is a maximal linearly ordered subset of $T$.\nAn Aronszajn tree is a tree with these three properties:\n\n1. Its height is $\\omega_1$, the least [[uncountable|0X55M9XET4]] ordinal: the levels $\\alpha \u003c \\omega_1$ are nonempty, and level $\\omega_1$ is empty.\n2. Each level is [[countable|0ZG61TFBES]].\n3. It has no branch of length $\\omega_1$, that is, no uncountable branch.\n\n## Remarks\n\nAronszajn trees exist in ZFC, by a construction of Nachman Aronszajn.\nOne construction uses bounded, strictly increasing sequences of rational numbers, each of countable successor length, ordered by extension.\nLevels are chosen by recursion on $\\alpha \u003c \\omega_1$, so that each level is countable and each node has extensions at all higher levels with only slightly larger supremum.\nAn uncountable branch would give a strictly increasing sequence of rationals of length $\\omega_1$, which is impossible since $\\mathbb{Q}$ is countable.\nThis shows that König's lemma, which gives an infinite branch in each infinite finitely branching tree, has no analogue at $\\omega_1$.\nAn Aronszajn tree in which every antichain is countable is a Suslin tree, and its existence is independent of ZFC.\n\n## Explanations\n\n### L1 human\n\nPicture a tree that grows upward through $\\omega_1$ levels, with only countably many nodes on each level.\nYou might expect some path to climb all the way to the top.\nAn Aronszajn tree is a tree of this kind in which every path stops at some countable level.\nIt shows that infinite combinatorics at $\\omega_1$ behaves very differently from the countable case.\n",
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